Rigidity of Carnot groups relative to multicontact structures

Mari, Filippo De; Ottazzi, Alessandro

doi:10.1090/s0002-9939-10-10212-3

Cited by 5 publications

(5 citation statements)

References 13 publications

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“…A closely related property is the so called rigidity property of quasiconformal or multicontact maps, also referred to as Liouville's property, but where the question is the finite dimensionality of the group of (locally defined) quasiconformal or multicontact maps, see [80], [74], [23], [68], [69], [65], [24], [56].…”

Section: The Riemannian Hessianmentioning

confidence: 99%

The Obata sphere theorems on a quaternionic contact manifold of dimension bigger than seven

Ivanov¹,

Petkov²,

Vassilev³

2017

J. Spectr. Theory

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We prove quaternionic contact versions of two of Obata's sphere theorems. On a compact quaternionic contact (qc) manifold of dimension bigger than seven and satisfying a Lichnerowicz type lower bound estimate we show that if the first positive eigenvalue of the sub-Laplacian takes the smallest possible value then, up to a homothety of the qc structure, the manifold is qc equivalent to the standard 3-Sasakian sphere. The same conclusion is shown to hold on a non-compact qc manifold which is complete with respect to the associated Riemannian metric assuming the existence of a function with traceless horizontal Hessian. The third result of the paper is a qc version of Liouville's theorem showing that a qc-conformal diffeomorphism between open connected sets of the 3-Sasakian sphere is a restriction of an element of the qc-conformal automorphism group of the sphere.

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Section: The Riemannian Hessianmentioning

confidence: 99%

The Obata sphere theorems on a quaternionic contact manifold of dimension bigger than seven

Ivanov¹,

Petkov²,

Vassilev³

2017

J. Spectr. Theory

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“…A closely related property is the so called rigidity property of quasiconformal or multicontact maps, also referred to as Liouville's property, but where the question is the finite dimensionality of the group of (locally defined) quasiconformal or multicontact maps, see [230], [199], [68], [192], [193], [185], [77], [163].…”

Section: The Qc Lichnerowicz Theoremmentioning

confidence: 99%

“…The definition of the "horizontal" space in the tangent bundle of the sphere and the distance function on the sphere require a few more details for which we refer to [67] and [14]. Multicontact maps and their rigidity in Carnot groups have been studied in [197], [199], [154], [70], [71], [41], [77], [192], [193], [194].…”

Section: The Cayley Transformmentioning

confidence: 99%

The Lichnerowicz and Obata first eigenvalue theorems and the Obata uniqueness result in the Yamabe problem on CR and quaternionic contact manifolds

Ivanov

Vassilev

2015

Nonlinear Analysis

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We report on some aspects and recent progress in certain problems in the sub-Riemannian CR and quaternionic contact (QC) geometries. The focus are the corresponding Yamabe problems on the round spheres, the Lichnerowicz-Obata first eigenvalue estimates, and the relation between these two problems. A motivation from the Riemannian case highlights new and old ideas which are then developed in the settings of Iwasawa sub-Riemannian geometries.

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“…In the noncommutative case, as is the situation we are interested in, the procedure was generalized by Tanaka [15] and it was used to generalize the study of infinitesimal automorphisms of G-structures by different authors. For the contact structures, the Tanaka prolongation theory was used in [18] and more recently by the authors of this article in different collaborations [4,9,10,17].…”

Section: Prolongation Of the Differential Equationsmentioning

confidence: 99%

“…The classical Liouville theorem states that C 4 -conformal maps between domains of R 3 are the restriction of the action of some element of the group O (1,4). The same result holds in R n when n > 3 (see, e.g., Nevanlinna [6]).…”

Section: Introductionmentioning

confidence: 99%

A Liouville Type Theorem for Carnot Groups: A Case Study

Ottazzi

Warhurst

2013

Trends in Harmonic Analysis

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In [1], the authors show that if φ is 1-quasiconformal on an open subset of a Carnot group G, then composition with φ preserves Q-harmonic functions, where Q denotes the homogeneous dimension of G. Then they combine this with a regularity theorem for Q-harmonic functions to show that φ is in fact C ∞ . As an application, they observe that a Liouville type theorem holds for some Carnot groups of step 2.In this article we argue, using the Engel group as an example, that a Liouville type theorem can be proved for every Carnot group. Indeed, the fact that 1-quasiconformal maps are smooth allows us to obtain a Liouville type theorem by applying the Tanaka prolongation theory [15].2000 Mathematics Subject Classification. 30L10, 20F18.

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Rigidity of Carnot groups relative to multicontact structures

Abstract: Abstract. We prove a rigidity type result for stratified nilpotent Lie algebras which gives a positive answer to a special case of a conjecture formulated by M. Cowling and of another conjecture formulated by A. Korányi.

Cited by 5 publications

References 13 publications

The Obata sphere theorems on a quaternionic contact manifold of dimension bigger than seven

The Obata sphere theorems on a quaternionic contact manifold of dimension bigger than seven

The Lichnerowicz and Obata first eigenvalue theorems and the Obata uniqueness result in the Yamabe problem on CR and quaternionic contact manifolds

A Liouville Type Theorem for Carnot Groups: A Case Study

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